Gade-Nielsen, Nicolai Fog

Online Research Database In Technology

                       General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.  • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal   If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.     Downloaded from orbit.dtu.dk on: Dec 18, 2017Accelerating interior point methods with GPUs for smart grid systemsGade-Nielsen, Nicolai FogPublication date:2011Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):Gade-Nielsen, N. F. (2011). Accelerating interior point methods with GPUs for smart grid systems [Sound/Visualproduction (digital)]. Workshop on "GPU Computing Today and Tomorrow", Kongens Lyngby, Denmark,01/01/2011, http://gpulab.imm.dtu.dk/courses.htmlD  T  U      I  n  f  o  r  m  a  t  i  c  sAccelerating interior point methods with GPUs forsmart grid systemsNicolai Fog Gade-NielsenD  T  U      I  n  f  o  r  m  a  t  i  c  sDepartment of Informatics and Mathematical ModellingTechnical University of DenmarkGPU Computing Today and Tomorrow, 2011Technical University of DenmarkNicolai Fog Gade-Nielsen (GPUlab) Accelerating IPMs with GPUs for SG systems DTU 2011-08-18 1 / 13D  T  U      I  n  f  o  r  m  a  t  i  c  sOutline1 Introduction2 Smart grid test case3 Interior point method4 Future plansNicolai Fog Gade-Nielsen (GPUlab) Accelerating IPMs with GPUs for SG systems DTU 2011-08-18 2 / 13D  T  U      I  n  f  o  r  m  a  t  i  c  sIntroductionIntroductionPh.D. student at GPUlab, DTU IMM since November 2010.Title: Scientific GPU Computing for Dynamical OptimizationInvestigate GPUs for solving optimization problems, primarily modelpredictive control (MPC).Initial test problem is distribution and control of electrical power viasmart grid through the use of MPC.Completed M.Sc. at DTU IMM in October 2010:Title: Implementation and evaluation of fast computational methodsfor high-resolution ODF problems on multi-core and many-coresystemsUsed GPUs for ray tracing and matrix-free SpMV.Nicolai Fog Gade-Nielsen (GPUlab) Accelerating IPMs with GPUs for SG systems DTU 2011-08-18 3 / 13D  T  U      I  n  f  o  r  m  a  t  i  c  sSmart grid test caseWhat is a smart grid?Dynamically control energy production and consumptionaccording to some objective, eg:Increased use of uncontrollable renewable power such as wind andsolar.Lower production costs.Lower CO2 production.etc.ExamplesHeat pump control in housesPower plant production controlNicolai Fog Gade-Nielsen (GPUlab) Accelerating IPMs with GPUs for SG systems DTU 2011-08-18 4 / 13D  T  U      I  n  f  o  r  m  a  t  i  c  sSmart grid test casePower plants in DenmarkImage credit: Economic MPC for Power Management in the Smart Grid | Tobias Gybel HovgaardNicolai Fog Gade-Nielsen (GPUlab) Accelerating IPMs with GPUs for SG systems DTU 2011-08-18 5 / 13D  T  U      I  n  f  o  r  m  a  t  i  c  sSmart grid test caseSimple Economic MPC 1Prediction horizon of N time steps.Np number of power plants with different properties.Properties on each power plant:Response time.Minimum and maximum change in one time step.Minimum and maximum power production.Cost.Simple example: Two power plants.A cheap but slow power plant.An expensive but fast power plant.Nicolai Fog Gade-Nielsen (GPUlab) Accelerating IPMs with GPUs for SG systems DTU 2011-08-18 6 / 13D  T  U      I  n  f  o  r  m  a  t  i  c  sSmart grid test caseSimple Economic MPC example0 50 100 150 200 250 300 350 400 450 50005101520Total Power  ProductionReference0 50 100 150 200 250 300 350 400 450 500051015P.G. #10 50 100 150 200 250 300 350 400 450 500051015P.G. #2Nicolai Fog Gade-Nielsen (GPUlab) Accelerating IPMs with GPUs for SG systems DTU 2011-08-18 7 / 13D  T  U      I  n  f  o  r  m  a  t  i  c  sSmart grid test caseSimple Economic MPC 2Define power plant production as an economic MPC problem:min φ = g′s + c′xs.t . Ax − s = bs ≥ 0c is the cost and x is the control variables of each power plant ineach time step.s is a slack variable and g is the cost of using it.A is our constraint matrix which encodes the constraints of eachpower plant and dynamics of the system. Very sparse and highlystructured.Number of control variables: Np ∗ N + NNicolai Fog Gade-Nielsen (GPUlab) Accelerating IPMs with GPUs for SG systems DTU 2011-08-18 8 / 13D  T  U      I  n  f  o  r  m  a  t  i  c  sInterior point methodIPM Algorithm (without Predictor-Corrector)While not converged doCompute duality gap.Set centering parameter.Compute residuals.Solve newton step. 0 −A′ 0−A 0 I0 S Λ∆x∆λ∆s = −rLrsrλ (1)[0 −A′−A −D−1] [∆x∆λ]= −[rLrs − Λ−1rλ](2)D = S−1ΛS = diag(s),Λ = diag(λ)Compute step length.Update step.Nicolai Fog Gade-Nielsen (GPUlab) Accelerating IPMs with GPUs for SG systems DTU 2011-08-18 9 / 13D  T  U      I  n  f  o  r  m  a  t  i  c  sInterior point methodSolving the Newton stepBottleneck: Solving the Newton stepNormally solved with direct method:Compute the Hessian matrix and use Cholesky factorization.HA = AT (D)ACholesky : HA = LLTImplemented using CUBLAS and MAGMA.What is MAGMA?’The MAGMA project aims to develop a dense linear algebra librarysimilar to LAPACK but for heterogeneous/hybrid architectures,starting with current "Multicore+GPU" systems.’Very fast Cholesky factorization.Developed at University of Tennessee.Nicolai Fog Gade-Nielsen (GPUlab) Accelerating IPMs with GPUs for SG systems DTU 2011-08-18 10 / 13D  T  U      I  n  f  o  r  m  a  t  i  c  sInterior point methodResults1500 2000 2500 3000510152025303540Control variablesSpeed−upSpeed−up of using GPUs vs sequential CPU (Xeon E5620)  Tesla C2070 (dense)Tesla C2070 (structured)GTX 590 (dense)GTX 590 (structured)Nicolai Fog Gade-Nielsen (GPUlab) Accelerating IPMs with GPUs for SG systems DTU 2011-08-18 11 / 13D  T  U      I  n  f  o  r  m  a  t  i  c  sInterior point methodProblemsIll-conditioningNormal equations -> condition number squaredHA = AT (D)ACholesky : HA = LLTIn the later iterations of the IPM, either si or λi goes toward 0. 0 −A′ 0−A 0 I0 S Λ∆x∆λ∆s = −rLrsrλ (1)[0 −A′−A −D−1] [∆x∆λ]= −[rLrs − Λ−1rλ](2)D = S−1ΛS = diag(s),Λ = diag(λ)Loss of sparsityNp = 1000,N = 48: 48048 decision variables.480482 hessian matrix to factorize: about 17 GB.Nicolai Fog Gade-Nielsen (GPUlab) Accelerating IPMs with GPUs for SG systems DTU 2011-08-18 12 / 13D  T  U      I  n  f  o  r  m  a  t  i  c  sFuture plansFuture plansIterative methods and preconditioningIdentify iterative solvers for solving the newton step iteratively.Identify good preconditioners.Evaluate which of the solvers and preconditioners are suitable forGPU implementation.Create toolbox for solving optimization problems using GPUsSmart grid problems used as test case for toolbox.Nicolai Fog Gade-Nielsen (GPUlab) Accelerating IPMs with GPUs for SG systems DTU 2011-08-18 13 / 13

Accelerating interior point methods with GPUs for smart grid systems

http://orbit.dtu.dk/ws/files/5854127/74D9Dd01.pdf

Accelerating interior point methods with GPUs for smart grid systems

Abstract

Similar works

Full text

Available Versions

Online Research Database In Technology